The 'Logical Problem of the Trinity', due to Richard Cartwright (1987), consists of identifying the following elements of the doctrine of the Trinity and asking how they can all be consistent:
(S1) The Father is God.
(S2) The Son is God.
(S3) The Holy Spirit is God.
(S4) The Father is not the Son.
(S5) The Father is not the Holy Spirit.
(S6) The Son is not the Holy Spirit.
(S7) There is exactly one God.
Call all of these together, 'P'. Branson notes that it is generally assumed (on analogy with the Logical Problem of Evil) that there is a wide range of different possible defenses of the consistency of P, and also that in fact that defenses have tended to cluster around two poles, Relative Identity and Social Trinitarianism. What Branson shows is that, while the exact form of the latter two may owe something to historical contingency, it is not surprising that versions of these two keep coming up, because positions at least approximately like them are the only positions that fulfill certain basic conditions.
There are two completely different ways people have argued that P is inconsistent. One, which Branson calls LPT-1, takes (S1)-(S3) to be identities ('God' as singular term), with (S7) using a counting schema with identity; the other, LPT-2, takes (S1)-(S3) to be ordinary predications ('God' as predicate nominative), with (S7) using a counting schema with predication. In either case, predicate logic with classical identity (PLI) will result in a contradiction. So addressing the Logical Problem of the Trinity requires establishing an interpretation of P under which neither LPT-1 nor LPT-2 is viable. All the solutions Branson looks at assume predicate logic is an adequate logic for this situation.* They also assume either the identity or the predication version of the standard analytic counting scheme.** Then there are only a few positions that have been proposed.
(1) Social (ST). On ST, (S1)-(S3) are not taken to involve identities, but predications, not (logically) any different from saying "Paul is human, Peter is human, James is human". This on its own is not quite adequate, since proponents of ST also tend to accept the analytic counting schema for (S7). But if we think of the parallel, Paul is human, Peter is human, James is human, you can accept a version of (S7), at least on some metaphysics of human nature, without getting an inconsistency, because the predicate in the counterpart to (S7) wouldn't mean exactly the same thing as it does in the counterparts to (S1), (S2), and (S3); it would mean one common human nature, as opposed to the individual human natures in (S1)-(S3). Thus ST also holds that the predicate 'God' in (S1)-(S3) does not mean exactly what it does in (S7). If that's the case, LPT-1 and LPT-2 both are inadequate due to not capturing the difference.
(2) Relative Identity (RI). Peter Geach famously noted an important problem in translating natural language identity statements into statements involving classical identity -- classical identity is 'absolute', but natural language identity statements are at least very often relativized to a kind. For instance, we say things like, "Mark Twain and Samuel Clemens are the same person", which is an identity statement, but one about identity of personhood. You can try to work around this using classical identity, but it gets increasingly complicated and controvertible. So, assuming that this is so because you ultimately can't adequately translate these relative identity statements by classical identity, there are two positions -- either there are two kinds of identity, relative and classical, or all identity is in reality relative. For the purposes of this discussion, it doesn't much matter which you choose (although you can get slightly different accounts of where LPT-1 in particular goes wrong depending on other assumptions). Thus the RI position, while it accepts predicate calculus as an appropriate for this topic, proposes that PLI is using the wrong account of identity; instead, you should use a version of the predicate calculus with relative identiy (RIL). Assessing this is complicated a bit by the fact that, since in PLI, classical identity is used in a lot of definitions (e.g., singular term reference), you have to relativize those, as well. But allowing for this, you can give a translation of P that uses identity all the way through and is consistent.
(3) Arianism. The Arian position on the Trinity is immune to the Logical Problem of the Trinity -- indeed, historically they obviously put forward arguments against the orthodox position that are forerunners of the Logical Problem of the Trinity. The Arians held that Father and Son both should be said to be God but not in the same sense, just as an animal and a picture of an animal can both be called 'animal', but not in the same sense. Thus LPT-1 and LPT-2 fail because there is an equivocation in (S1), (S2), and (S3).
(4) Naive Modalism (NM). The Sabellian or modalist position (or at least what is often assumed to be the Sabellian position) on the Trinity is also sometimes proposed. On this view, (S4), (S5), and (S7) are not non-identities, and therefore we don't get an inconsistency. As Branson notes, this requires some potentially implausible fudging about how we interpret these three components of P -- in the most plausible interpretations, you are simply denying them. But it's a position that has been proposed that avoids the problem.
As Branson goes on to argue, while you could have variations on each of these, something like these are the only real options for handling the Logical Problem of the Trinity. When you abstract from details and look at how each position works in the abstract, there are only a few positions possible.***
(I) LPT-1 or LPT-2
The doctrine of the Trinity is logically inconsistent.
PLI is not an appropriate logic for analyzing this subject, due to an inadequate account of identity. In another, more adequate logic, translation of P yields no contradiction. RI is the primary historical form this has taken in analytic philosophy.
(S4)-(S6) are at least not true in the sense in which they would have to be understood to generate the contradiction; that is, however they are to be understood, they are not non-identities.
There is an equivocation in (S1)-(S3), and therefore P involves no contradiction. This is the Arian position.
There is an equivocation between (S1)-(S3) on the one hand and (S7) on the other, and therefore P involves no contradiction. ST is the primary historical form this has taken in analytic philosophy.
(I), (III), and (IV) are automatically ruled out by the orthodox doctrine of the Trinity, since they are respectively the heresies of Unitarianism, Sabellianism, and Arianism. Thus, given the background assumptions of analytic philosophy, the only ways to defend orthodox Trinitarianism from the Logical Problem of the Trinity are (II) and (V). This explains why the discussion has stabilized around the two poles of RI and ST -- while details of particular positions may be due to other things, something like these are the only non-heretical options. (Of course, you could accept both. This is not a popular option, which would be a matter worth investigating.****) Branson ends:
The Trinitarian speculations of philosophers might help with the metaphysics of the Trinity, with establishing the Biblical basis for it, or with some rhetorical or other issue. But from a purely formal point of view, they will always be just another member of one of the Families of answers to the LPT we have defined here, and will necessarily share the controversial features that define those families.
* This is not a trivial assumption; predicate logic arguably presupposes that you are applying it to things with an at-least-in-principle precisely discrete extensions, which is a worrisome presupposition in the context of the Trinity. Predicate calculus, after all, was developed largely to talk about sets, which are entirely constituted by elements that are already assumed to be discrete, and applying it to anything messier than that often requires elaborate workarounds. In addition, predicate calculus can't easily handle things like modes of predication in scholastic term logic, which have historically been used in this context. But the vast majority of analytic philosophers treat predicate calculus as the one true, or at least the fundamental, logical system, and therefore the assumption that predicate logic is applicable is almost universal among analytic philosophers of religion.
** Also not a trivial assumption; it is not difficult to find Church Fathers like St. Basil or scholastics like St. Thomas denying that (S7) should be a counting statement at all -- in Platonic and Aristotelian systems of philosophy, there are more fundamental senses of 'one' than the kind you can count. But analytic philosophy historically developed, at least in great measure, out of philosophy of mathematics, and thus the assumption that a statement like (S7) is a counting statement is also going to be almost universal among analytic philosophers of religion.
*** I simplify Branson's actual taxonomy here; he has reasons for going a more complicated road, but they aren't relevant to the overall point.
**** My own guess is that the sharp divide is actually due to the anti-Trinitarian positions: people don't find LPT-1 and LPT-2 to be equally plausible objections (because they interpret key claims differently), so if they think LPT-1 is more plausible, they gravitate to (II), and then (V) looks like unnecessary interpretation-gerrymandering; if they think LPT-2 more plausible, they gravitate to (V), and then (II) looks like unnecessary logic-chopping. That is, people are not looking at the whole abstract field of arguments (which can only in any case be understood after a lot of arguing has already happened); they are addressing the objections they themselves think are most serious and that influences the responses that they think are most serious.